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Simplifying 24x2 + 140x + -100 = 0 Reorder the terms: -100 + 140x + 24x2 = 0 Solving -100 + 140x + 24x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(-25 + 35x + 6x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-25 + 35x + 6x2)' equal to zero and attempt to solve: Simplifying -25 + 35x + 6x2 = 0 Solving -25 + 35x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.166666667 + 5.833333333x + x2 = 0 Move the constant term to the right: Add '4.166666667' to each side of the equation. -4.166666667 + 5.833333333x + 4.166666667 + x2 = 0 + 4.166666667 Reorder the terms: -4.166666667 + 4.166666667 + 5.833333333x + x2 = 0 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + 5.833333333x + x2 = 0 + 4.166666667 5.833333333x + x2 = 0 + 4.166666667 Combine like terms: 0 + 4.166666667 = 4.166666667 5.833333333x + x2 = 4.166666667 The x term is 5.833333333x. Take half its coefficient (2.916666667). Square it (8.506944446) and add it to both sides. Add '8.506944446' to each side of the equation. 5.833333333x + 8.506944446 + x2 = 4.166666667 + 8.506944446 Reorder the terms: 8.506944446 + 5.833333333x + x2 = 4.166666667 + 8.506944446 Combine like terms: 4.166666667 + 8.506944446 = 12.673611113 8.506944446 + 5.833333333x + x2 = 12.673611113 Factor a perfect square on the left side: (x + 2.916666667)(x + 2.916666667) = 12.673611113 Calculate the square root of the right side: 3.560001561 Break this problem into two subproblems by setting (x + 2.916666667) equal to 3.560001561 and -3.560001561.Subproblem 1
x + 2.916666667 = 3.560001561 Simplifying x + 2.916666667 = 3.560001561 Reorder the terms: 2.916666667 + x = 3.560001561 Solving 2.916666667 + x = 3.560001561 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.916666667' to each side of the equation. 2.916666667 + -2.916666667 + x = 3.560001561 + -2.916666667 Combine like terms: 2.916666667 + -2.916666667 = 0.000000000 0.000000000 + x = 3.560001561 + -2.916666667 x = 3.560001561 + -2.916666667 Combine like terms: 3.560001561 + -2.916666667 = 0.643334894 x = 0.643334894 Simplifying x = 0.643334894Subproblem 2
x + 2.916666667 = -3.560001561 Simplifying x + 2.916666667 = -3.560001561 Reorder the terms: 2.916666667 + x = -3.560001561 Solving 2.916666667 + x = -3.560001561 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.916666667' to each side of the equation. 2.916666667 + -2.916666667 + x = -3.560001561 + -2.916666667 Combine like terms: 2.916666667 + -2.916666667 = 0.000000000 0.000000000 + x = -3.560001561 + -2.916666667 x = -3.560001561 + -2.916666667 Combine like terms: -3.560001561 + -2.916666667 = -6.476668228 x = -6.476668228 Simplifying x = -6.476668228Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.643334894, -6.476668228}Solution
x = {0.643334894, -6.476668228}
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